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An integral involving Levy process with no positive jumps

Let $L_t$ be a Levy process that has no positive jumps, but is not strictly decreasing, i.e $$ L_t = \gamma t + \sigma B_t + J_t, $$ where $B_t$ is a Brownian motion, $J_t$ is a pure jump process with ...
bm76's user avatar
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101 views

A dependent and discrete version of the Komlós-Major-Tusnády theorem

The well-known Komlós-Major-Tusnády approximation gives sharp speed of convergence of a uniform empirical process to a Brownian bridge. Here I am considering how to approach a similar problem with ...
John Wong's user avatar
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2 votes
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126 views

A question related to the jumps of a Levy process

The Lévy–Khintchine formula says that any Lévy process, $X=(X(t), t \geq 0)$, has a specific form for its characteristic function. More precisely, for all $t \geq 0$, $u \in \mathbb R^d$: $$ \mathbb{E}...
André Goulart's user avatar
2 votes
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128 views

Proof of the Lévy–Itō decomposition in this paper

Let $E$ be a normed $\mathbb R$-vector space; $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$; $(X_t)_{t\ge0}$ be an $E$...
0xbadf00d's user avatar
  • 167
2 votes
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96 views

Invariant measures of Levy S.D.Es

Suppose we call a real valued stochastic process $\{Z_t\}$ to be distributed as ${\cal S}\alpha{\cal S}(\sigma)$ if each of the characteristic functions is $\phi_{Z_t}(u) = \exp\left\{-t\vert \sigma u ...
gradstudent's user avatar
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2 votes
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146 views

Modulus of continuity of Lévy process as jump size tends to zero

While reading Kallenberg's "Foundations of Modern Probability Theory", 2nd edition, the following question regarding an argument in the proof of Lemma 15.19 occurred to me. Let $X_n(t)$ be a sequence ...
Uchiha's user avatar
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2 votes
0 answers
278 views

Radon-Nikodym for continuous time processes

Likelihood theory for statistical inference concerning stochastic processes in continuous time are well used. How ever i've found no real literature concerning the fundamentals. What is know from ...
ziT's user avatar
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31 views

$\alpha$ stable processes without jumps

Levy processes with jumps can be formulated following the Levy-kinchkine representation, which provide a decomposition of the characteristic function into three factors corresponding to the diffusion (...
user1172131's user avatar
1 vote
0 answers
58 views

Drift of reverse SDE with Lévy processes ($\alpha$ stable distributions)

Given an SDE with a Lévy process with a drift $b(x,t)$ the reverse SDE will have a drift, $\tilde{b}(x,t)$, given by the relation: $$\tilde{b}(x,t) = - b(x,t) + \int_{\mathbb{R}} y \left( 1 + \frac{...
user1172131's user avatar
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140 views

Ask assistance for finding K. Sato - Lévy Processes on the Euclidean Spaces

The paper me and my professor want is called K. Sato (1995) Lévy Processes on the Euclidean Spaces, Lecture Notes, Institute of Mathematics, University of Zurich. I tried to find the paper on the ...
Zoël Li's user avatar
1 vote
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175 views

Interpretation of the Lévy measure of an infinitely divisible random vector

We know that a random vector $X$ is infinitely divisible (ID) if for all $n \in \mathbb N$, there exist $X_1^n,..., X_{n}^n$ i.i.d. random vectors such that: \begin{equation} X = X_1^n + ...+ X_n^...
PSE's user avatar
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142 views

What are the Lévy processes with specific increments?

It is known that the increment of the Wiener process $W$ is drawn from a Gaussian distribution, i.e. $\Delta W \sim \mathcal{N}(0, \delta t)$. I wonder what are the Lévy processes with increments from ...
user482699's user avatar
1 vote
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47 views

How do we need to argue in this step of the Itō-Lévy-Khintchine decomposition?

Let $E$ be a $\mathbb R$-Banach space; $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$; $(X_t)_{t\ge0}$ be an $E$-...
0xbadf00d's user avatar
  • 167
1 vote
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191 views

Characterization of Poisson random measure in terms of Laplace transform

Let $(E,\mathcal E)$ be a measurable space and $\mu$ be a measure on $(E,\mathcal E)$. A random measure $\pi$ on $(E,\mathcal E)$ is called Poisson with intensity $\mu$ if $\pi(B)\sim\operatorname{...
0xbadf00d's user avatar
  • 167
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36 views

Interpretation of Lévy process with signed Lévy measures

Suppose that I have a non-decreasing, pure jump Lévy process of finite variation $X$ with Lévy measure $\pi$. The Lévy measure is then supported on $(0,+\infty)$. Suppose that the Lévy measure is a ...
NancyBoy's user avatar
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Step in the derivation of the total idle time distribution of an M/G/1 queue

I'm trying to work my way through the proof of Thm. 1.11 in Kyprianou's Introductory Lectures on Fluctuations of Levy Processes with Applications but really struggle to understand the following step. ...
Othman El Hammouchi's user avatar
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74 views

What is the sufficient and necessary condition for Blumenthal-Gettor index = 0?

This question comes from the following paper 1961(Blumenthal) Let us consider a Levy process $X$ whose Levy triplet is $(a,s,\nu)$. According the above paper, Blumenthal-Gettor index is given by $$\...
Fractional analysics's user avatar
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150 views

Define the convolution root of probability measures on a measurable group

Let $(G,\mathcal G)$ be a measurable group and $\nu^{\ast k}$ denote the $k$th convolution power of a probability measure $\nu$ on $(G,\mathcal G)$ for $k\in\mathbb N$. Remember that a probability ...
0xbadf00d's user avatar
  • 167
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62 views

Hitting order of sets by a Lévy process

Let $X$ be a transient Lévy process on $\mathbb R$, and $B\subseteq \mathbb R$ a Borel set with first hitting time $T_B = \inf \left\{t>0 : X_t\in B\right\}$. For Borel $A\subseteq B$, can anything ...
user1118's user avatar